In the previous labs, you dropped stuff and let them fall. While falling, you assumed that the objects had only the gravitational force acting on them. This isn’t quite true. When objects move through the air, there is also an air resistance force. One model for the air resistance says it has a magnitude of:
Where ρ is the density of the air, A is the cross sectional area, C is a drag coefficient that depends on shape and v is the magnitude of the velocity of the object.
In this lab, you will:
- Verify that this model of air drag works for falling coffee filters
- Find the AC term for falling coffee filters
- Model the motion (with vpython) of a falling coffee filter and compare it to real data.
So, to start – here is a video showing the details in air resistance.
Thinking about air resistance
Before starting the experiment, think about the following questions:
1) Suppose I take a coffee filter and release it from rest.
– What will be its initial speed?
– What will be an expression for the gravitational force?
– What will be its acceleration?
– Draw a free-body diagram for this object at this instant
2) Now suppose the coffee filter has fallen for 0.1 seconds. How will the answers for the above questions change?
3) What will happen to the air resistance force as it falls? What will happen to the acceleration as it falls?
4) Eventually, the filter will reach a state of equilibrium. When it is going fast enough that the air resistance force is equal to the weight, the free body diagram will look like this:
What the acceleration be at this time? (zero). This is called terminal velocity (because it will no longer speed up)
There is a reason that coffee filters are used. They have a nice property of being stackable. If I drop one coffee filter, or two stacked together, the shape and size does not change. Only the weight changes. This allows us to drop a different number of filters and see how they fall.
1. Set up the motion detector to measure motion in the vertical direction.
2. Start the motion detector and drop the coffee filter (wait until the detector is actually taking data).
3. Obtain the terminal velocity for this drop. Remember, the terminal velocity is when the acceleration is zero (you will actually never get here, but you can get close enough). If you are using the position-time graph, you can select a portion of the graph and fit a quadratic equation to this data. If the coefficient of the squared term is near zero, this is a good selection for terminal velocity. You can then fit a linear function to find the terminal velocity.
If you use a velocity-time graph, you can fit a linear function to a sub-set of data to determine the acceleration. You can then read the value of the velocity from the graph.
3a. Also record the height the filter drops and the time it takes to fall (you will use this to verify your numerical model).
4. Repeat 4 times to get an average terminal velocity.
5. Repeat the above for 1, 2, 3, 4, and 5 filters.
6. Determine the mass of one of the filters (assume they all have the same mass).
7. Create a graph of terminal velocity squared versus mass. From this graph, you can find the slope which should be related to the coefficient “c”
Modeling the motion
In the last lab, you made a program for a falling ball. You might want start with that program. Hopefully, you had something like this in the beginning:
ball=sphere(pos=(0,1.2,0), radius=(0.01)) g=vector(0,-9.8,0) ball.m = 0.4
And then later, in your loop you calculated the force (or really you could have calculated this before the loop since it didn’t change):
Fnet = ball.m*g
Now, this no longer works. Now you have a force that changes as the ball speeds up. This means that you have to include this force inside the loop. The above force code will work if you add in the air resistance. Here are some tips:
- If you want, you can assume the ball is only falling down. What would this mean about the vector direction of the air resistance force?
- If you don’t treat your air resistance as a vector, bad things will happen
- If you want your program to be even more useful, you will have the program determine the direction of the air resistance force. This means that when the object is moving up, the force is down and when the object moves down, the force is up.
Here are some useful vpython tips. If I have a vector (say the momentum of the ball), I can find the magnitude and the unit vector for that momentum like this:
ball.p=vector(1.3,-3.2,0) p = mag(ball.p) #this is the magnitude of a vector phat = norm(ball.p) #this is a unit vector in the direction of ball.p
Some other things to consider: how small should your time interval be (deltat?). In general, the smaller the better. But too small will just make things run slow. So try this. Pick a value for deltat and run the program. Now make deltat half the size and run it again. If you get about the same value at the end (for say the final velocity and for the time it takes to fall) then the bigger deltat was ok.
How do you know if your calculations are even correct? You have to make a connection to real life. How do you do that? Well, you have these falling coffee filters. Save the data from one of those runs. It doesn’t matter which one, but you need to know the mass of the coffee filters.
Now use those same exact values in your vpython program. You should be able to reproduce similar values for final velocity and the time it takes fall a certain distance.